Generally the best (in the least squares sense in the linear Helmholtz
function space)
coefficient vector able to represent a true
(domain-normalized) eigenstate
has exponentially large coefficients.
This can be expressed (following Vergini [194])
in terms of a basis `badness',

(J.2)

which diverges exponentially once the basis set size is increased
beyond .
Numerically, because of the null-space truncation of Section 5.3.2,
has an upper limit of
.
This issue is identical to the divergence of basis coefficients found
by Dietz et al. when attempting to expand a `cake billiard' eigenfunction in terms
of plane waves (or angular-momentum states) [63].
The problem is that of representing evanescent components, which are
generally present in billiard eigenfunctions, by distributions
of real plane waves.
As Berry has shown [26], rapidly-oscillating distributions
are required which grow exponentially with the number of
wavelengths across the system and with the evanescence parameter.
This was well understood by Vergini, who demonstrated that
improving the basis by adding a small number of evanescent plane waves
(in the case of the stadium only)
enabled to reach .
This improved the basis tension error
from about
to about .

In general the requirements of a good basis are small
and, if possible,
for most states.
I first give the real plane waves,
which work well in many billiard shapes,
then give Vergini's evanescent plane
waves which currently are well-adapted only to the stadium billiard (and shapes
close to it).